Mut'a marriage is a type of marriage in Islam in which the time of the marriage is agreed between man and woman and Mahr is paid to the woman (Dowry) just like permanent marriage, but there is no financial responsibility of expenses or housing for the woman on the man. It ends with the end of the agreed time or even before that if the man leaves the remaining period. It can be extended after the end or termination of its time and there will be no Iddah (waiting period) if it is between same same man and same woman.
Mut'a marriage can be done with a Muslim of Ahlul Kitab woman who is not currently married nor passing IDDAH which is the waiting period after previous marriage (around three months) or death or husband (Four months and ten days).
Mut'a marriage like the permanent marriage, needs recitation of AQD (agreement) in Arabic between the woman and the man either by themselves or by proxy on their behalf.
More details are available on the websites of Islamic Rules.